heat and mass transfer analysis for the mhd flow of ... · heat and mass transfer analysis for the...

Ain Shams Engineering Journal (2016) xxx, xxx–xxx

Ain Shams University

Ain Shams Engineering Journal

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ENGINEERING PHYSICS AND MATHEMATICS

Heat and mass transfer analysis for the MHD

flow of nanofluid with radiation absorption

* Corresponding author.

E-mail addresses: [email protected] (P. Durga Prasad), kksaisiva@

gmail.com (R.V.M.S.S. Kiran Kumar), [email protected]

(S.V.K. Varma).

Peer review under responsibility of Ain Shams University.

Production and hosting by Elsevier

http://dx.doi.org/10.1016/j.asej.2016.04.0162090-4479 � 2016 Faculty of Engineering, Ain Shams University. Production and hosting by Elsevier B.V.This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Please cite this article in press as: Durga Prasad P et al., Heat and mass transfer analysis for the MHD flow of nanofluid with radiation absorption, Ain Sham(2016), http://dx.doi.org/10.1016/j.asej.2016.04.016

P. Durga Prasad *, R.V.M.S.S. Kiran Kumar, S.V.K. Varma

Department of Mathematics, S.V. University, Tirupati 517502, A.P, India

Received 12 October 2015; revised 1 April 2016; accepted 29 April 2016

KEYWORDS

MHD;

Chemical reaction;

Dufour effect;

Nanofluid;

Porous medium

Abstract In this paper the effects of Diffusion thermo, radiation absorption and chemical reaction

on MHD free convective heat and mass transfer flow of a nanofluid bounded by a semi-infinite flat

plate are analyzed. The plate is moved with a constant velocity U0, temperature and the concentra-

tion are assumed to be fluctuating with time harmonically from a constant mean at the plate. The

analytical solutions of the boundary layer equations are assumed of oscillatory type and are solved

by using the small perturbation technique. Two types of nanofluids namely Cu-water nanofluid and

TiO2-water nanofluid are used. The effects of various fluid flow parameters are discussed through

graphs and tables. It is observed that the diffusion thermo parameter/radiation absorption param-

eter enhance the velocity, temperature and skin friction. This enhancement is very significant for

copper nanoparticles. This is due to the high conductivity of the solid particles of Cu than those

of TiO2. Also it is noticed that the solutal boundary layer thickness decreases with an increase in

chemical reaction parameter. It is because chemical molecular diffusivity reduces for higher values

of Kr.� 2016 Faculty of Engineering, Ain Shams University. Production and hosting by Elsevier B.V. This is an

open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

It is well known that the convectional heat transfer fluids such

as water, mineral oil and ethylene glycol have poor heat trans-fer properties compared to those of most solids in general. An

innovative way of improving the heat transfer of fluids is tosuspend small solid particles in the fluids. This new kind of

fluid named as nanofluid was first introduced in 1995 by Choi[1]. The term nanofluid is used to describe a solid liquidmixture which consists of basic low volume fraction of highconductivity solid nanoparticles.

A nanofluid is a fluid in which nanometer-sized particles aresuspended in a convectional heat transfer fluid to improve theheat transfer characteristics. Thus, nanofluids have many

applications in industry such as coolants, lubricants, heatexchangers and micro-channel heat sinks. Therefore, numer-ous methods have been taken to improve the thermal conduc-

tivity of these fluids by suspending nano/micro sized particlematerials in liquids. They reported breakthrough in substan-tially increasing the thermal conductivity of fluids by adding

s Eng J

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2 P. Durga Prasad et al.

very small amounts of suspended metallic oxide nanoparticles(Cu, CuO and Al2O3) to the fluid [1–3].

In recent years, the natural convection flow of nanofluid

has been studied by [4,5]. Kuznetsov and Nield [6] investigatedthe natural convective boundary-layer flow of a nanofluid pasta vertical plate using Buongiorno model. Gorla and Chamkha

[7] studied the natural convective boundary layer flow of nano-fluid in a porous medium. Convective heat transfer and flowcharacteristics of nanofluids are given by [8,9].

Magnetohydrodynamic boundary-layer flow of nanofluidand heat transfer has received a lot of attention in the fieldof several industrial, scientific, and engineering applicationsin recent years. Nanofluids have many applications in indus-

tries, since materials of nanometer size with unique chemicaland physical properties have sundry applications such as elec-tronics cooling, and transformer cooling, and this study is

more important in industries such as hot rolling, melt spinning,extrusion, glass fiber production, wire drawing, manufactureof plastic and rubber sheets, and polymer sheet and filaments.

In view of the abovementioned applications of nanofluids,many researchers contributed in this area. Model and compar-ative study for peristaltic transport of water based nanofluids

was given by Shehzad et al. [10]. Ghaly [11] considered thethermal radiation effects on a steady flow, whereas Raptisand Massalas [12] and El-Aziz [13] have analyzed the unsteadycase. Mutuku-Njane and Makinde [14] investigated the hydro-

magnetic boundary layer flow of nanofluids over a permeablemoving surface with Newtonian heating. Takhar et al. [15]have studied the radiation effects on the MHD free convection

flow of a gas past a semi-infinite vertical plate. The naturalconvective boundary layer flows of a nanofluid past a verticalplate have been described by Kuznetsov and Nield [16,17]. In

this model, the Brownian motion and thermophoresis areaccounted with the simplest possible boundary conditions.Bachok et al. [18] have shown the steady boundary-layer flow

of a nanofluid past a moving semi-infinite flat plate in a uni-form free stream. It was assumed that the plate is moving inthe same or opposite directions to the free stream to definethe resulting system of non-linear ordinary differential equa-

tions. Raptis and Kafousis [19] have investigated steady hydro-dynamic free convection flow through a porous mediumbounded by an infinite vertical plate with constant suction

velocity. Raptis [20] discussed unsteady two-dimensional natu-ral convection flow of an electrically conducting, viscous andincompressible fluid along an infinite vertical plate embedded

in a porous medium. A comprehensive survey of convectivetransport in nanofluids was made by Buongiorno [21]. Alsohe gave an explanation for the abnormal increases of the ther-mal conductivity of nanofluids after examining many mechan-

ics in the absence of turbulence and he found that theBrownian diffusion and the thermophoresis effects are themost important and reported conservation laws for nanofluids

in the presence of these two effects.Magnetohydrodynamic stagnation-point flow of a power-

law fluid toward a stretching surface in the presence of thermal

radiation and suction/injection was studied by Mahapatraet al. [22]. MHD mixed convection in a nanofluid due to astretching/shrinking surface with suction/injection and heat

and mass transfer of nanofluid through an impulsively verticalstretching surface using the spectral relaxation method wasstudied by Haroun et al. [23,24]. The Radiation effects on anunsteady MHD axisymmetric stagnation-point flow over a

Please cite this article in press as: Durga Prasad P et al., Heat and mass transfer analy(2016), http://dx.doi.org/10.1016/j.asej.2016.04.016

shrinking sheet in the presence of temperature dependent ther-mal conductivity with Navier slip were considered by Mondalet al. [25]. Hamad and Pop [26] investigated an unsteady MHD

free convective flow of nanofluid past a vertical permeable flatplate with constant heat source. MHD stagnation point flowand heat transfer problem from a stretching sheet in the pres-

ence of a heat source/sink and suction/injection in porousmedia was studied by Mondal et al. [27]. In their study, it isseen that spectral perturbation method can be used to find

numerical solutions for complicated expansions encounteredin perturbation schemes. Turkyilmazoglu [28,29] analyzedthe heat and mass transfer effects in MHD flow of nanofluids.

Chemical reaction effects on heat and mass transfer are of

considerable importance in hydrometallurgical industries andchemical technology. Several investigators have examined theeffect of chemical reaction on the flow, heat and mass transfer

past a vertical plate. Convective boundary-layer flow of MHDNanofluid over a stretching surface with chemical reactionusing the spectral relaxation method was studied by Haroun

et al. [30]. Venkateswarlu and Satyanarayana [31] have studiedthe effects of chemical reaction and radiation absorption onthe heat and mass transfer flow of nanofluid in a rotating sys-

tem. Recently, Kiran Kumar et al. [32,33] analyzed the studyof heat and mass transfer enhancement in free convection flowwith chemical reaction and thermo-diffusion in nanofluidsthrough porous medium in a rotating frame.

In the abovestated papers, the diffusion-thermo andthermal-diffusion terms were neglected from the energy andconcentration equations respectively. But when heat and mass

transfer occurs simultaneously in a moving fluid, the relationbetween the fluxes and the driving potentials is of intricate nat-ure. It has been found that an energy flux can be generated not

only by temperature gradient but also by concentration gradi-ents. The energy flux caused by concentration gradient is calledthe Dufour or diffusion-thermo effect. The diffusion-thermo

(Dufour) effect was found to be of considerable magnitudesuch that it cannot be ignored by Eckert and Darke [34]. Inview of the importance of this diffusion-thermo effect, Jhaand Singh [35] studied the free convection and mass transfer

flow about an infinite vertical flat plate moving impulsivelyin its own plane.

Motivated by some of the researchers mentioned above and

its applications in various fields of science and technology, it isof interest to discuss and analyze the Diffusion thermo andchemical reaction effects on the free convection heat and mass

transfer flow of nanofluid over a vertical plate embedded in aporous medium in the presence of radiation absorption andconstant heat source under fluctuating boundary conditions.Majority of the studies, reported in the literature, on free con-

vective heat and mass transfer in nanofluid embedded in a por-ous medium deal with local similarity solutions and non-similarity solutions. But in the present study, the governing

equations are solved using similarity transformations.

2. Formulation of the problem

An unsteady natural convectional flow of a nanofluid past avertical permeable semi-infinite moving plate with constantheat source is considered. The physical model of the fluid flow

is shown in Fig. 1. The flow is assumed to be in the x-directionwhich is taken along the plate and y-direction is normal to it.

sis for the MHD flow of nanofluid with radiation absorption, Ain Shams Eng J


Figure 1 Schematic diagram of the physical problem.

Heat and mass transfer analysis for MHD flow 3

A uniform external magnetic field of strength B0 is taken tobe acting along the y-direction. It is assumed that the inducedmagnetic field and the external electric field due to the polar-

ization of charges are negligible. The plate and the fluid are

at the same temperature T01 and concentration C0

1 in a sta-

tionary condition, when t P 0, the temperature and concentra-tion at the plate fluctuate with time harmonically from a

constant mean. The fluid is a water based nanofluid containingtwo types of nanoparticles either Cu (copper) or TiO2 (Tita-nium oxide). The nanoparticles are assumed to have a uniform

shape and size. Moreover, it is assumed that both the fluidphase nanoparticles are in thermal equilibrium state. Due tosemi-infinite plate surface assumption, furthermore the flow

variables are functions of y and time t only.Under the above boundary layer approximations, the gov-

erning equations for the nanofluid flow are given by

@v0

@y0¼ 0 ð1Þ

qnf

@u0

@t0þ v0

@u0

@y0

� �¼ lnf

@2u0

@y02þ ðqbÞnfgðT0 � T0

1Þ �lnfu

0

K0 � rB20u

0

ð2Þ

@T0

@t0þ v0

@T0

@y0

� �¼ anf

@2T0

@y02� Q0

ðqCpÞnfðT0 � T0

1Þ þQ0lðC0 � C0

1Þ

þ DmKT

CsðqCpÞnf@2C0

@y02ð3Þ

@C0

@t0þ v0

@C0

@y0¼ DB

@2C0

@y02� KlðC0 � C0

1Þ ð4Þ

where u0 and v0 are the velocity components along x and y axesrespectively. bnf is the coefficient of thermal expansion of nano-

fluid, r is the electric conductivity of the fluid, qnf is the density

of the nanofluid, lnf is the viscosity of the nanofluid, qCp

� �nfis

the heat capacitance of the nanofluid fluid, g is the acceleration

due to gravity, K0 is the permeability porous medium, T0 is thetemperature of the nanofluid, Q is the temperature dependent

volumetric rate of the heat source, and anf is the thermal diffu-

sivity of the nanofluid, which are defined as follows [36], where/ is the solid volume fraction of the nanoparticles, Knf and Ks

are thermal conductivities of the base fluid and of the solidrespectively. The thermo-physical properties of the pure fluid(water), copper and titanium which were used for code valida-

tion are given in Table 1.The boundary conditions for the problem are given by

t0 < 0; u0ðy0; t0Þ ¼ 0; T0 ¼ T01; C0 ¼ C0

1t0 P 0; u0ðy0; t0Þ ¼ U0; T0 ¼ T0

w þ ðT0w � T0

1Þeeiw0t0 ; C0 ¼ C0w þ ðC0

w � C01Þeeiw

0t0 at y0 ¼ 0

u0ðy0; t0Þ ¼ 0; T0 ¼ T01; C0 ¼ C0

1asy0 ! 1ð5Þ

where T0 is the local temperature of the nanofluid and Q is theadditional heat source. On the other hand, bf and bC are the

coefficients of thermal expansion of the fluid and of the solid,respectively, qf and qC are the densities of the fluid and of the


solid fractions, respectively, while qnf is the viscosity of the

nanofluid, anf is the thermal diffusivity of the nanofluid, and

ðqCpÞnf is the heat capacitance of the fluid, which are defined

as (Abbasi [37,38])

qnf ¼ ð1� /Þqf þ /qs; ðqCpÞnf ¼ ð1� /ÞðqCpÞf þ /ðqCpÞs;ðqbÞnf ¼ ð1� /ÞðqbÞf þ /ðqbÞs;

Knf ¼ Kf

Ks þ 2Kf � 2/ðKf � KsÞKs þ 2Kf þ 2/ðKf � KsÞ� �

; lnf ¼lf

ð1� /Þ2:5 ;

anf ¼ Knf

ðqCpÞnfð6Þ

v0 ¼ �V0 ð7Þwhere the constant �V0 represents the normal velocity at theplate which is positive suction ðV0 > 0Þ and negative forblowing injection ðV0 < 0Þ.

Let us introduce the following dimensionless variables:

u ¼ u0

U0

; y ¼ U0y0

mf; t ¼ U2

0t0

mf; x ¼ mfx0

U20

;

h ¼ ðT0 � T01Þ

ðT0w � T0

1Þ; S ¼ V0

U0

; M ¼ rB20mf

qfU20

Du ¼ DmKTðC0w � C0

1ÞkfCsðT0

w � T01Þ ; QL ¼ Q0

lðC0w � C0

1ÞU2

0ðT0w � T0

1Þ;

Kr ¼ KlmfU2

0

; Sc ¼ mfDB

Q ¼ Q0m2fKfU

20

; Pr ¼ mfaf; K ¼ K0qfU

20

m2f;

Gr ¼ ðqbÞfgmfðT0w � T0

1ÞqfU

30

; w ¼ ðC0 � C01Þ

ðC0w � C0

1Þ ð8Þ

Here Pr ¼ vfafis the Prandtl number, S is the suction ðS > 0Þ or

injection ðS < 0Þ parameter,M is the Magnetic parameter, andQL is the radiation absorption parameter, Kr is the chemicalreaction parameter, Sc is the Schmidt number, Gr is the



Table 1 Thermo-physical properties (see [36,10]).

Physical properties Water Copper (Cu) Titanium

oxide (TiO2)

Cp ðJ=kg KÞ 4179 385 686.2

q ðkg=m3Þ 997.1 8933 4250

k ðW=m KÞ 0.613 400 8.9538

b� 10�5 ð1=KÞ 21 1.67 0.9


Grashof number, K is the permeability parameter, and Du isthe Diffusion-thermo parameter.

In Eq. (2)–(4) with the boundary conditions (5), we get

A@u

@t� S

@u

@y

� �¼ D

@2u

@y2þ BGrh� Mþ 1

K

� �u ¼ 0 ð9Þ

C@h@t

� S@h@y

�QLw

� �¼ 1

PrE@2h@y2

�Qh

� �þDu

Pr

@2w@y2

ð10Þ

0 1 20

0.5

1

1.5

2

2.5

3

3.5

4

4.5

u

S=0.1,0.2,0.3

Cu Water

0 1 20

0.5

1

1.5

2

2.5

3

3.5

4

4.5

u

S=0.1,0.2,0.3

TiO2 Water

(a)

(b)

Figure 2 (a) and (b) Velocity profiles for S with Sc ¼ 0:60;Kr ¼


@w@t

� S@w@y

¼ 1

Sc

@2w@y2

� Krw ð11Þ

With the boundary conditions

t < 0 : u ¼ 0; h ¼ 0; w ¼ 0

t P 0 : u ¼ 1; h ¼ 1þ eeixt; w ¼ 1þ eeixt at y ¼ 0

u ¼ 0; h ¼ 0; w ¼ 0 as y ! 1ð12Þ

3. Solution of the problem

Eqs. (9)–(11) are coupled non-linear partial differential equa-tions whose solutions in closed-form are difficult to obtain.

To solve these equations by converting into ordinary differen-tial equations, the unsteady flow is superimposed on the meansteady flow, so that in the neighborhood of the plate, the

expressions for velocity, temperature and concentration areassumed as

uðy; tÞ ¼ u0 þ eu1eixt

3 4 5 6y

Solid Line: Pure fluidDotted Line: Nano fluid (φ =0.25)

3 4 5 6y

Solid Line: Pure fluidDotted Line:Nano fluid (φ =0.25)

0:5;Q ¼ 2;Du ¼ 0:5;K ¼ 4;/ ¼ 0:25;M ¼ 0:5;QL ¼ 2;Gr ¼ 2.



0 1 2 3 4 5 60

0.5

1

1.5

2

2.5

3

3.5

4

4.5

y

u

Cu Water

Du=1,2,3

Solid Line: Pure fluidDotted Line: Nano fluid(φ =0.20)

0 1 2 3 4 5 60

0.5

1

1.5

2

2.5

3

3.5

4

4.5

y

u

TiO2 Water

Du=1,2,3

Solid Line: Pure fluidDotted Lie: Nano fluid (φ=0.20)

(a)

(b)

Figure 3 (a) and (b) Velocity profiles for Du with Sc ¼ 0:60;S ¼ 0:1;Kr ¼ 0:5;Q ¼ 2;K ¼ 4;/ ¼ 0:20;M ¼ 0:5;QL ¼ 2;Gr ¼ 2.


hðy; tÞ ¼ h0 þ eh1eixt

wðy; tÞ ¼ w0 þ ew1eixt ð13Þ

where eð� 1Þ is a parameter.Eqs. (9)–(11) are reduced to

Du000 þ ASu00 � Mþ 1

K

� �u0 ¼ �BGrh0 ð14Þ

Du001 þ ASu01 � Mþ 1

K

� �þ Aix

� �u1 ¼ �BGrh1 ð15Þ

Eh000 þ PrCSh00 �Qh0 ¼ �Duw000 � PrCQLw0 ð16Þ

Eh001 þ PrCSh01 � ðQþ PrCixÞh1 ¼ �Duw001 � PrCQLw1 ð17Þ

w000 þ SScw0

0 � KrScw0 ¼ 0 ð18Þ


w001 þ SScw0

1 � ðixþ KrÞScw1 ¼ 0 ð19ÞThe boundary conditions (12) become

u0 ¼ 1; u1 ¼ 0; h0 ¼ 1; h1 ¼ 1; w0 ¼ 1; w1 ¼ 1 at y¼ 0

u0 ¼ 0; u1 ¼ 0; h0 ¼ 0; h1 ¼ 0; w0 ¼ 0; w1 ¼ 0 as y!1ð20Þ

By solving Eqs. (14)–(19) using the boundary conditions (20),we get

uðy; tÞ ¼ B5e�m5y þ B3e

�m3y þ B4e�m1yð Þ

þ e B8e�m6y þ B6e

�m4y þ B7e�m2yð Þeixt ð21Þ

hðy; tÞ ¼ B1e�m3y þ A1e

�m1yð Þ þ e B2e�m4y þ A2e

�m2yð Þeixt ð22Þ

wðy; tÞ ¼ e�m1yð Þ þ e e�m2yð Þeixt ð23ÞThe shearing stress at the plate in dimensional form is given

by



0 1 2 3 4 5 60

0.5

1

1.5

2

2.5

3

3.5

y

uQL=1,2,3

Cu water Solid Line: Pure fluidDotted Line:Nano fluid (φ =0.05)

0 1 2 3 4 5 60

0.5

1

1.5

2

2.5

3

3.5

y

u QL=1,2,3

TiO2 Water Solid Line: Pure fluidDotte Line: Nano fluid (φ =0.05)

(a)

(b)

Figure 4 (a) and (b) Velocity profiles for QL with Sc ¼ 0:60;S ¼ 0:1;Kr ¼ 0:1;Q ¼ 2;Du ¼ 2;K ¼ 5;/ ¼ 0:05;M ¼ 0:5;Gr ¼ 2.


s ¼ @u

@t

� �y¼0

¼ ð�B5m5 � B3m3 � B4m1Þ

þ eð�B8m6 � B6m4 � B7m2Þeixt ð24Þ

Similarly, the rate of heat transfer at the plate/Nusselt num-ber is given by

Nu ¼ � @h@t

� �y¼0

¼ ðB1m3 þ A1m1Þ þ eðB2m4 þ A2m2Þeixt

ð25Þ

The rate of mass transfer at the plate/Sherwood number is

given by

Sh ¼ � @w@t

� �y¼0

¼ m1 þ em2eixt ð26Þ


4. Results and discussions

In order to bring out the salient features of the flow, heat andmass transfer characteristics with nanoparticles, the results arepresented in Figs. 2–10 and in Tables 2 and 3. The effects ofnanoparticles on the velocity, the temperature and the concen-

tration distributions as well as on the skin friction and the rateof heat and transfer coefficients are discussed numerically. Wehave chosen here e ¼ 0:02; t ¼ 1; x ¼ 1, and Pr ¼ 0:71, whilethe remaining parameters are varied over a range, which arelisted in figures.

4.1. Effect of suction parameter ðSÞ

Fig. 2(a) and (b) demonstrates the effect of suction parameterS on fluid velocity u for both regular ð/ ¼ 0Þ and nanofluid

ð/–0Þ. As an output of figures, it is seen that the velocity of



0 1 2 3 4 5 60

0.5

1

1.5

2

2.5

3

3.5

y

u

M=0.2,0.4,0.6

Cu Water Solid Line: Pure fluidDotted Line: Nano fluid (φ =0.05)

0 1 2 3 4 5 60

0.5

1

1.5

2

2.5

3

3.5

y

u

M=0.2,0.4,0.6

TiO2 Water Solid Line: Pure fluidDotted Line: Nano fluid (φ =0.05)

(b)

(a)

Figure 5 (a) and (b) Velocity profiles for M with Sc ¼ 0:60;S ¼ 0:1;Kr ¼ 0:5;Q ¼ 2;Du ¼ 2;K ¼ 5;/ ¼ 0:05;QL ¼ 1;Gr ¼ 2.


the fluid across the boundary layer decreases by increasing the

suction parameter S for both regular fluid and nanofluid withnanoparticles Cu and TiO2. It is worth mentioned here that theinfluence of the suction parameter S on the fluid velocity is

more effective for nanofluid with the nanoparticles Cu andTiO2. It is also observed that the maximum velocity of Cu–wa-ter nanofluid is higher than that of the TiO2–water nanofluidattains in the neighborhood of y ¼ 1. Fig. 9 displays the effects

of the suction parameter ðSÞ on the species concentration pro-files. As the suction parameter increases the species concentra-tion, the solutal boundary layer thickness decreases. This is

due to the usual fact that the suction stabilizes the boundarygrowth. These consequences are obviously supported fromthe physical point of view.

4.2. Effect of Dufour number (Du)

The influence of Diffusion-thermo parameter ðDuÞ on thevelocity distribution for Cu–water and TiO2–water


nanofluids is shown in Fig. 3(a) and (b) respectively. It is

noticed that the boundary layer thickness increases withDu for both regular and nanofluids. Thus the hydrodynamicboundary layer thickness increases as the Dufour number

increases. Fig. 6(a) and (b) depicts the effects of Du on thetemperature profiles within the boundary layer. With theincreasing values of Du, the temperature of nanofluid isfound to increase for both Cu and TiO2 nanofluids, i.e.,

Du causes to increase the thermal boundary layer thickness.Also the thermal boundary layer decreases faster for lowervalues of Dufour number for both regular fluid and nanoflu-

ids. Physically, Decreasing Du clearly reduces the influenceof species gradients on the temperature field, so that temper-ature function values are clearly lowered and the boundary

layer regime is cooled. On other hand concentration func-tion in the boundary layer regime is increased as Du isdecreased. Mass diffusion is evidently enhanced in thedomain as a result of the contribution of temperature

gradients.



0 1 2 3 4 5 60

0.5

1

1.5

2

2.5

3

y

θ

Du=1,2,3


0 1 2 3 4 5 60

0.5

1

1.5

y

θ

Du=1,2,3

TiO2 Water Solid Line: Pure fluidDotted Line: Nano fluid (φ =0.05)

(b)

(a)

Figure 6 (a) and (b) Temperature profiles for Du with Sc ¼ 0:60;S ¼ 0:1;Kr ¼ 2;Q ¼ 2;/ ¼ 0:05;QL ¼ 2;Gr ¼ 2.


4.3. Effect of radiation absorption parameter ðQLÞ

Figs. 4(a), (b) and 7(a), (b) are graphical representation of thevelocity and temperature profiles for different values ofQL with

Cu and TiO2 nanoparticles. It is clear from these figures that thevelocity and temperature profiles increase with increase of QL.This is due to the fact that, when heat is absorbed, the buoyancy

force accelerates the flow. Also, it is observed that in the case ofCu-nanoparticles thermal boundary layer is very thicker thanthat of TiO2-nanoparticles. The large QL values correspond

to an increased dominance of conduction over absorption radi-ation thereby increasing buoyancy force and thickness of thethermal and momentum boundary layers.

4.4. Effect of magnetic field parameter ðMÞ

Fig. 5(a) and (b) presents the typical nanofluid velocity profilesfor various values of magnetic field parameter ðMÞ for the


nanoparticles Cu and TiO2. From these graphs, it is obvious

that the nanofluid velocity of the fluid decelerates with anincrease in the strength of magnetic field. The effects of a trans-verse magnetic field on an electrically, conducting fluid give

rise to a resistive-type force called the Lorentz force. This forcehas the tendency to slow down the motion of the fluid in theboundary layer. These results quantitatively agree with theexpectations, since magnetic field exerts retarding force on nat-

ural convection flow. Also, it is clear that the nanofluid veloc-ity is lower for the regular fluid and the velocity reaches themaximum peak value for the nanoparticle TiO2 in the compar-

ison of Cu nanoparticles.

4.5. Effect of Schmidt number ðScÞ

The variation in the concentration boundary layer of the flowfield is shown in Fig. 8 for H2;H2O vapor and NH3. This figuredepicts the concentration distribution in the presence of the



0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

y

θ

QL=1,2,3


0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

1.2

1.4

y

θ

QL=1,2,3

TiO2 Water Solid Line: Pure fluidDotted Line: Nano fluid (φ=0.05)

(a)

(b)

Figure 7 (a) and (b) Temperature profiles for QL with Sc ¼ 0:60;S ¼ 0:1;Kr ¼ 2;Q ¼ 2;/ ¼ 0:05;Gr ¼ 2;Du ¼ 2.


flow field. Comparing the curves of the said figure, it isobserved that the growing Schmidt number decreases the con-

centration boundary layer thickness of the flow field at allpoints. This causes the concentration buoyancy effects todecrease yielding a reduction in the fluid flow. The reductionsin the concentration profiles are accompanied by simultaneous

reductions in the concentration boundary layer.

4.6. Effect of chemical reaction parameter ðKrÞ

For different values of destructive chemical reaction parameterKrð> 0Þ, the concentration profiles are plotted in Fig. 10. Anincrease in chemical reaction parameter will suppress the

concentration of the fluid. Higher values of Kr amount to a fall


in the chemical molecular diffusivity, i.e., less diffusion.Therefore, they are obtained by species transfer. An increase

in Kr will suppress species concentration. The concentrationdistribution decreases at all points of the flow field with theincrease in the reaction parameter.

The numerical values of the Skin-friction coefficient and

Nusselt number for the nanoparticles Cu and TiO2 are pre-sented in Tables 2 and 3. From Table 2, it is seen that thelocal Nusselt number decreases with increasing values of

suction parameter S, Dufour number Du, and volume frac-tion parameter /, while it increases with increasing values ofradiation absorption parameter QL and heat source parame-

ter Q for both the nanoparticles Cu and TiO2. It is alsoobserved that the rate of heat transfer is higher in the



0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

1.4

y

ψ

Sc=0.22 (Hydrogen)Sc=0.30 (Helium)Sc=0.60 (Water Vapour)Sc=0.78 (Ammonia)

Figure 8 Concentration profiles for Sc with S ¼ 2;Kr ¼ 0:1.

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

1.4

y

ψ

S=1S=2S=3S=4

Figure 9 Concentration profiles for S with Sc ¼ 0:60;Kr ¼ 0:1.


Cu–water nanofluid than in TiO2–water nanofluid. This is

due to the high conductivity of the solid particles Cu thanthose of TiO2. From Table 3 it is clear that the local skinfriction coefficient increases with the increasing values ofGr;K;QL;Du;Kr and /, as magnetic field creates Lorentz

force which decreases the value of skin friction for boththe nanoparticles Cu and TiO2.

5. Conclusions

In the present study, we have theoretically studied the effectsof the metallic nanoparticles on the unsteady MHD free con-

vective flow of an incompressible fluid past a moving infinite


vertical porous plate. The set of governing equations are

solved analytically by using perturbation technique. Theeffects of various fluid flow parameters on velocity, tempera-ture and species concentration, Skin-friction and the rate ofheat transfer coefficient are derived and discussed through

graphs and tables. The following conclusions are made fromthe present investigation:

1. In the boundary layer region, fluid velocity decreases withthe increasing values of magnetic field parameter and suc-tion parameter for both the nanoparticles Cu and TiO2,

while it increases with the increasing values of Dufour num-ber and radiation absorption parameter.



0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

1.4

y

ψ

Kr=1Kr=2Kr=3Kr=4

Figure 10 Concentration profiles for S with Sc ¼ 0:60;S ¼ 1.

Table 2 Numerical values of Skin-friction coefficient ðCfÞwith Sc ¼ 0:60;S ¼ 0:1; t ¼ 1;Q ¼ 2;Pr ¼ 0:71.

Gr M K QL Du Kr / Skin

friction for

copper

water fluid

Skin friction

for titanium

oxide water

fluid

1 0.5 2 1.5 0.5 0.5 0.15 0.1161 0.3000

2 0.5 2 1.5 0.5 0.5 0.15 1.9215 1.5062

3 0.5 2 1.5 0.5 0.5 0.15 3.9550 3.3080

2 0.1 2 1.5 0.5 0.5 0.15 4.2564 4.2403

2 0.5 2 1.5 0.5 0.5 0.15 1.9215 1.5062

2 1.0 2 1.5 0.5 0.5 0.15 0.4719 0.0533

2 0.5 2 1.5 0.5 0.5 0.15 1.9215 1.5062

2 0.5 4 1.5 0.5 0.5 0.15 3.0118 2.6953

2 0.5 6 1.5 0.5 0.5 0.15 3.5858 3.3745

2 0.5 2 1 0.5 0.5 0.15 0.7440 0.4441

2 0.5 2 3 0.5 0.5 0.15 5.4544 4.6951

2 0.5 2 5 0.5 0.5 0.15 10.1650 8.9472

2 0.5 2 1.5 0.1 0.5 0.15 1.4828 1.1090

2 0.5 2 1.5 0.2 0.5 0.15 1.5925 1.2083

2 0.5 2 1.5 0.3 0.5 0.15 1.7021 1.3076

2 0.5 2 1.5 0.5 0.1 0.15 0.0279 0.2332

2 0.5 2 1.5 0.5 5 0.15 2.0418 2.1227

2 0.5 2 1.5 0.5 10 0.15 2.2124 2.2715

2 0.5 2 1.5 0.5 0.5 0.05 0.9921 0.8832

2 0.5 2 1.5 0.5 0.5 0.15 1.9215 1.5062

2 0.5 2 1.5 0.5 0.5 0.2 2.8177 2.0827

Table 3 Numerical values of Nusselt number

ðNuÞSc ¼ 0:60; t ¼ 1;Pr ¼ 0:71.

S Du / QL Q Nu for copper

water fluid

Nu for titanium

oxide water fluid

0.1 0.5 0.15 1.5 2 0.7928 0.8416

0.5 0.5 0.15 1.5 2 0.7224 0.8307

1.0 0.5 0.15 1.5 2 0.4004 0.6923

0.1 0.1 0.15 1.5 2 0.9221 0.9555

0.1 0.2 0.15 1.5 2 0.8898 0.9270

0.1 0.3 0.15 1.5 2 0.8574 0.8986

0.1 0.5 0.05 1.5 2 0.8381 0.8415

0.1 0.5 0.15 1.5 2 0.7928 0.8416

0.1 0.5 0.2 1.5 2 0.6688 0.8023

0.1 0.5 0.15 2 2 0.4558 0.5502

0.1 0.5 0.15 4 2 0.8920 0.6157

0.1 0.5 0.15 6 2 2.2399 1.7815

0.1 0.5 0.15 1.5 3 0.9468 1.4675

0.1 0.5 0.15 1.5 4 1.4974 1.9287

0.1 0.5 0.15 1.5 5 2.4122 2.3096


2. An increase in the Dufour number and radiation absorp-tion parameter leads to increase the thermal boundary layerthickness.

3. The species concentration decreases with the increasing val-

ues of suction parameter, chemical reaction parameter andSchmidt number.

4. There is a significant effect of M and QL on the skin friction

coefficient.


Appendix A

A ¼ ð1� /Þ þ /qs

qf

� �� ; B ¼ ð1� /Þ þ /

ðqbÞsðqbÞf

! !;

C ¼ ð1� /Þ þ /ðqCpÞsðqCpÞf

! !;

E ¼ knfkf¼

ð1þ 2/Þ þ ð2� 2/Þ Kf

Ks

� �ð1� 2/Þ þ ð2þ 2/Þ Kf

Ks

� �0@

1A; D ¼ 1

ð1� /Þ2:5 ;

A1 ¼ � m21Duþ PrQLC

Em21 � PrCSm1 �Q

� �;




A2 ¼ � m22Duþ PrQLC

Em22 � PrCSm2 � ðQþ ixPrCÞ

� �;

B1 ¼ ð1� A1Þ; B2 ¼ ð1� A2Þ;

B3 ¼ � BB1Gr

Dm23 � ASm3 � ðMþ 1

KÞ

!;

B4 ¼ � BA1Gr

Dm21 � ASm1 � ðMþ 1

KÞ

!; B5 ¼ ð1� B3 � B4Þ;

B6 ¼ � BB2Gr

Dm24 � ASm4 � ðAixþMþ 1

KÞ

!;

B7 ¼ � BA2Gr

Dm22 � ASm2 � ðAixþMþ 1

KÞ

!;

B8 ¼ �ðB6 þ B7Þ;m1 ¼SScþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðSScÞ2 þ 4KrSc

q2

0@

1A;

m2 ¼SScþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðSScÞ2 þ 4Scðixþ KrÞ

q2

0@

1A;

m3 ¼PrCSþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðPrCSÞ2 þ 4EQ

q2

0@

1A;

m4 ¼PrCSþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðPrCSÞ2 þ 4EðQþ ixPrCÞ

q2E

0@

1A;

m5 ¼ASþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðASÞ2 þ 4D Mþ 1

K

� �q2D

0@

1A;

m6 ¼ASþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðASÞ2 þ 4DðAixþ ðMþ 1

KÞÞ

q2D

0@

1A

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P. Durga Prasad is currently doing M.Phil. in

Sri Venkatewswara University, Tirupati,

Andhra Pradesh-517502, India. His area of

interest is FLUID DYNAMICS. He has

2 years research experience in research and

3 years of teaching experience at various

levels. He presented/attended several Interna-

tional/National conferences/seminars/work-s

hops.


R.V.M.S.S. Kiran Kumar is currently doing

Ph.D in Srivenketeswara University, Tirupati,

A.P.-517502, India. He is doing his Ph.D in

the area of FLUID DYNAMICS, and he has

2 years research experience and 2 years of

teaching experience at various levels. He is

qualified CSIR-NET With AIR-44, GATE-

2014 with AIR-281, GATE-2013 with AIR-

527 and awarded UGC MERITORIOUS

BSR-FELLOWSHIP. He is a life member of

Andhra Pradesh Mathematical Society.

Professor S.V.K. Varma, a senior professor in

the Department of Mathematics, Sri Venka-

teswara University, Tirupati, Andhra Pra-

desh, India. He has vast experience in teaching

and administration and also in research. His

area of research is Fluid Dynamics, Magneto

hydrodynamics, and Heat and Mass transfer.

He published 160 papers in national and

international journals. He collaborated with

foreign researchers such as A.J. Chamkha of

Kuwait, and J. Prakash of Botswana. He

presented several papers in various confer-

ences at national and international levels. He also visited abroad as a

resource person for an international conference at University of

Botswana, Gaborone, in 2013. He has organized several national

conferences and now is going to organize an international conference

with the collaboration of University of Botswana in June 2014. He

guided 15 students for Ph.D. and 18 for M.Phil. He is the life member

of various bodies. He is the member of Board of Studies of various

universities and Autonomous Institutions.

































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